Comments on: “Bootstrap Inference for Group Factor Models”

It is my pleasure to discuss the paper by Sílvia Gonçalves, Julia Koh, and Benoit Perron (henceforth GKP) presented as the 2024 Hal White Lecture at the 2024 SoFiE conference. The authors consider the group factor models studied, among others, by Andreou et al. (2019, 2020). GKP address the challenges of implementing the testing procedure for the number of common factors among groups outlined in Andreou et al. (2019) (henceforth AGGR). The challenges pertain to the nonstandard convergence rates and the presence of a bias correction appearing in the asymptotic distribution of the test statistic, both caused by the multi-step testing procedure. GKP propose and theoretically justify a simple bootstrap test that avoids the need to explicitly estimate the bias and variance of the canonical correlations used to identify the number of common factors. They provide high-level conditions and verify these conditions for a wild bootstrap scheme similar to one proposed by Gonçalves and Perron (2014). The GKP paper also covers an extension of the wild bootstrap scheme that is robust to serial and cross-sectional dependence of the idiosyncratic error terms. Simulations reported by GKP show that even under the same setting as in AGGR, the asymptotic distribution leads to size distortions in finite samples, while the wild bootstrap has nominal size under the null hypothesis. They also show that the extension of the bootstrap procedure leads to rejection rates closer to the nominal level compared to the asymptotic framework.

Once my co-authors and I became aware of the GKP paper showing the size distortions resulting from using the asymptotic bias corrections derived by AGGR, we adopted their wild bootstrap procedure in our own research, see Andreou et al. (2022, 2024).

GKP re-derived the asymptotic analysis of AGGR using regularity conditions amenable to establishing the validity of their bootstrap procedure. I would like to discuss some of the regularity conditions. For the asymptotic validity of the bootstrap, GKP require $E|eti|32<M$ and $E‖hti‖32$< M for $i=1,2$⁠. This amounts to assuming Gaussianity of the idiosyncratic errors, which is restrictive for many applications, particularly those involving panels of financial returns. Along the same lines, AGGR assume $N/T5/2→$0, where N = $min{N1,N2},$ with $Ni$ the cross-sectional sample sizes of the two panels and T the time series dimension. Instead, GKP make the more restrictive assumption that $N/T3/2$  $→$ 0, which simplifies the asymptotic expansions. This more restrictive assumption also infringes on applications involving financial returns. For example, Andreou et al. (2024) expand the original AGGR setting and assume a common component in the panel of idiosyncratic volatilities, which is relevant for many financial applications. Compared to the case without a common component in idiosyncratic volatilities, we have an additional term in the asymptotic variance as well as an additional bias correction term. By assuming N = $O(T2),$  Andreou et al. (2024) control the estimation error for the bias term at order $T−2,$ leaving only an additional term in the variance of the asymptotic distribution. While this falls outside of the GKP framework for the validity of the bootstrap, it would be interesting to explore extensions of the GKP bootstrap to establish asymptotic validity of their procedure to a broader class of models. It is of course challenging to deal with asymptotic expansions emerging from the general AGGR framework and adapt these to make them suitable for bootstrap statistics. What GKP achieved in their paper is already a major improvement of the approach pursued by AGGR.

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